Symmetry in a constrained Hamiltonian system with singular higher-order Lagrangian

Abstract

For a canonical formalism with a higher-order derivative, the corresponding generalized first Noether theorem (GFNT) for a constrained Hamiltonian system and the generalized Noether identities (GNI) for a system with a non-invariant action integral are derived, which may be useful to analyse the Dirac constraint for such a system. Using the GFNT another example is given in which Dirac's conjecture fails; using the GNI the strong and weak conservation laws are deduced and it is pointed out that for certain variant systems there is also a Dirac constraint. Suppose that there are only first-class constraints (FCC) in a system, then an algorithm for the construction of a gauge generator is developed, once the Hamiltonian and the FCC of the system with a higher-order Lagrangian are given.